“Mathematical coupling occurs when one variable directly or indirectly contains the whole or part of another, and the two variables are then analysed using correlation or regression. As a result, the statistical procedure of testing the null hypothesis — that the coefficient of correlation or the slope of regression is zero — might no longer be appropriate”

Regression model: \(\text{NCPV}_{1} - \text{NCPV}_0 = \alpha + \beta \, \text{NCPV}_0 + \varepsilon\)

The regression coefficient (slope): \(\beta = \frac{\operatorname{Cov}((\text{NCPV}_1 - \text{NCPV}_0),\ \text{NCPV}_0)}{\operatorname{Var}(\text{NCPV}_0)} = \frac{\operatorname{Cov}(\Delta \text{NCPV},\ \text{NCPV}_0)}{\operatorname{Var}(\text{NCPV}_0)}\)

This simplifies to: \(\beta = \frac{\rho\, \sigma_1 - \sigma_0}{\sigma_0}\)

where:

• \(\rho = \operatorname{Cor}(\text{NCPV}_0,\ \text{NCPV}_1)\)
  
• \(\sigma_0\), \(\sigma_1\) = SDs at baseline and follow-up

With baseline NCPV contributing to both predictor and outcome, the slope captures an inseparable combination of mathematical coupling and possible biological change.
It is not a clean estimate of baseline influence.

You cannot determine whether a positive or negative slope from ΔNCPV ∼ NCPV₀ reflects biology or math, because the math builds the relationship. To isolate biological effects, you must model follow-up directly with baseline as a covariate — not as part of the outcome.

“Statisticians have repeatedly warned against correlating/regressing change with baseline due to two methodological concerns known as mathematical coupling and regression to the mean.”

“Mathematical coupling can lead to an artificially inflated association between initial value and change score when correlation or regression is used.”

See also:
Analysis of ‘change scores’
Assessing the Relationship between the Baseline Value of a Continuous Variable and Subsequent Change Over Time
Mathematic coupling of data: a common source of error Revisiting the relation between change and initial value: a review and evaluation

“Estimated lifetime LDL-C exposure was only a significant predictor of final NCPV in the univariable analysis but lost significance when age was included as a covariate. Both age and lifetime LDL-C exposure lost significance when baseline CAC was included in the model.”

Frequentist:

Assuming there is no true association between ApoB and ΔNCPV, how likely is it that we’d observe a slope as large (or larger) than the one we found, just by chance?

If p-value (p > 0.05) a frequentist analysis can say:

“We did not find sufficient evidence to reject the hypothesis that ApoB has no association with ΔNCPV”

It cannot say the null is likely true, or produce the probability that there is no association, just that the data were inconclusive.

Bayesian:

How well do the data fit under two competing models, one with no association (null), and one with a range of plausible effect sizes for ApoB (alternative)?

A Bayes factor (e.g., BF₁₀ = 6) allows a stronger statement:

“The observed data are 6 times more likely (moderate evidence) under the ‘no association’ model than under the alternative model that assumes some effect from ApoB (as defined by the prior).”

• Frequentist: “No evidence of effect”
  
• Bayesian: “Evidence for no effect”

“Since lack of statistical signifcance (ie, P > 0.05) should not be interpreted as evidence in favor of the null but simply a failure to reject the null, the addition of Bayesian inference adds credence to finding that there is no association between NCPV vs LDL-C or ApoB…”

“Bayes factors were calculated using BayesFactor::regressionBF… and an ~ rscale value of 0.8 to contrast a moderately informative prior with a conservative distribution width (to allow for potential large effect sizes) due to the well-documented association between ApoB changes and coronary plaque changes”

Bayesian Prior: represents your belief about likely effect sizes before seeing the data.

From the BayesFactor documentation for the parameter rscaleCont:

“Several named values are recognized: ‘medium’, ‘wide’, and ‘ultrawide’, which correspond to rscales of √2/4, 1/2, and √2/2, respectively.”

• “medium” → rscale = √2 / 4 ≈ 0.354
• “wide” → rscale = 0.5
• “ultrawide” → rscale = √2 / 2 ≈ 0.707

rscale of 0.8 is wider than “ultrawide”. It is not a “moderately informative” prior. It’s actually a weakly informative or vague prior, placing most of its weight on large effects.

A moderately informative prior would typically correspond to “medium” (≈ 0.354) or “wide” (0.5), which place more mass on smaller effects.

“it is always important to conduct a prior sensitivity analysis to fully understand the influence that the prior settings have on posterior estimates”

“a researcher can have a very strong opinion about the model parameter values, and this opinion (via the prior) can drive the final model estimates.”

This kind of rscale sensitivity analysis is standard for default Bayes factors, but it’s a limited diagnostic — it tests only prior width, not prior plausibility or model fit.

Table 1. A heuristic classification scheme for Bayes factors BF₁₀ Source: SpringerLink

1. ΔNCPV as outcome
   Regressed (NCPV₁ − NCPV₀) on baseline NCPV₀
   Mathematical coupling → regression slopes reflect algebra, not just biology

2. Univariable regressions
   Each predictor tested separately (ApoB, LDL-C, age…)
   No confounder adjustment → biased, unreliable, low credibility estimates

3. Bayesian inference
   Bayes inference used to support “no ApoB effect” & “plaque begets plaque”
   Unadjusted, observational data → misleading, unusual use of Bayesian inference

4. Prior choice (rscale = 0.8)
   Prior assumes large effects
   No sensitivity analysis → results likely prior-driven

5. Bayes factor interpretation
   Claimed null is “6–10× more likely”
   Bayes factor misstated as posterior probability → compares model fit, not truth

6. Headline claim
   “Plaque Begets Plaque, ApoB Does Not”
   Overstates evidence → mathematically coupled, confounded, and fragile analysis

Baseline NCPV median = 44 mm³; TPS median = 0 → ≥50% of values are zero
CCTA cannot report negative plaque → both outcomes are left-censored at 0

This affects not just modeling but measurement:
When true plaque ≈ 0, error is asymmetric — it can only overestimate.

ΔNCPV, their primary outcome, is a change score between two bounded, skewed measures.
Likely to produce non-normal residuals and heteroscedasticity (e.g., larger spread at higher baseline).

If smaller baseline values were also linked to larger increases,
this may reflect the effects of left-censoring and error asymmetry — not true biological acceleration.

These issues are clear in TPS, and may affect NCPV, but diagnostics are not shown.

OLS assumes homoscedastic, normal residuals
performance::check_model() was run — but no output provided

They could have considered methods to address this such as: Tobit regression, log-transform, or robust SEs

• Study Registered Primary endpoint: %Δ NCPV over 12 mo, not specifically sized for ApoB detection.
• With n = 100, 80 % power is reached for large and medium effects for 1 predictor linear regression.
• A null result after one year may be due to low power, not proof of no ApoB effect.
• Note: even if it is adequately powered, it doesn’t negate all the other issues like mathematical coupling, confounding, non-adjustment, short follow-up etc.

“It should be emphasized that this includes heterogeneity in progression (and regression) across the population.” - Keta-CTA study

“If, despite our results show that CVDrisk among LMHRs is heterogeneous (and thus, a pooled summary isn’t a good idea), you must have a numerical pooled NCPVchange value, it is: p50=18.8 mm3 IQR(37.3).” - X post from Author

They say the group is heterogeneous (to downplay the pooled NCPV change), yet they ran univariable regressions and interpreted pooled Bayes factors as if the group were homogeneous.

If a group is too heterogeneous to report pooled outcomes, it is also too heterogeneous to justify pooled inferences about predictors or mechanisms.

If their CVD risk (e.g., plaque progression) is not coherent, then the category fails as a predictive or explanatory tool.

“p50=18.8 mm3 IQR(37.3).”

With a median of 18.8 mm³ and only 1–2 individuals showing regression, the IQR of 37.3 mm³ must be mostly skewed upward, not balanced.

A wide IQR here reflects high inter-individual variability in outcomes among the LMHRs, variability in progression of NCPV, as all but a few individuals had more plaque at follow up.

In other words: With nearly all LMHRs showing plaque progression, a wide IQR (37.3 mm³) doesn’t indicate balanced variability—it reflects differing degrees of worsening.

This matters because:

• The outcome (ΔNCPV) shows high variability across individuals in the LMHR group.
• That inflates standard errors, weakens statistical power, and makes small observed effects more ambiguous.
• It calls for adjusted modeling to reduce noise and account for confounding, not pooled univariable regressions and Bayes factors interpreted as strong evidence.

Letter to the Editor

Response to the Letter

From the response:

“Regarding the analytical points brought forward, we are aware of the relevance of linear assumptions to obtain accurate estimators. Since residual plot evaluation can also be subjective”

Objective, quantitative statistical tests also exist for assessing model assumptions:

• Normality: Shapiro-Wilk, Kolmogorov–Smirnov, Q-Q plots (visual / quantitative)
• Homoscedasticity: Breusch–Pagan, White test
• Influence / leverage: Cook’s distance, leverage plots (visual based on quantitative values)

The performance::check_model() function they cite automatically generates these checks.

It’s unclear why they wouldn’t include or reference the output.

“…we followed their suggestion and re-ran all models with robust linear regression…as expected, there were small differences with the published estimates, all models using robust regression were consistent with what was reported.”

No output, diagnostics, or model fit provided. We are asked to trust their assertion.

“We agree that being able to identify patients with rapid plaque progression, and gaining better understanding of the mechanisms that mediate its pace (i.e. insulin resistance, inflammation, different dietary composition elements, etc.) is paramount. We plan to address these risk factors in future reports”

Are they admitting the current analysis lacks adjustment for likely confounders, despite drawing strong conclusions from univariable regressions?

Just run the multivariable models.

“Moreover, our results are compatible with a causal role of ApoB in atherosclerosis, as we have openly acknowledged and supported in previous publications.”

“Plaque Begets Plaque, ApoB does not”

“Along the same lines, we would like to clarify that our title was not meant to be a statement about causality. “Plaque begets plaque” (which, of course, mirrors the proverb “Money begets money”) is frequently used to highlight the strong and clinically relevant association of baseline plaque values with plaque progression rate[7]. In retrospect, we might have chosen “Longitudinal Data from the KETO-CTA Study” as alternative phrasing to avoid misinterpretations.”

“misinterpretations”